**5.5.5. Total masses and mass distributions in clusters-the hydrostatic
method**

Masses for individual galaxies or for clusters of galaxies can be
derived from
the distribution of their X-ray emitting gas if this gas is in hydrostatic
equilibrium. This is a reasonable approximation as long as the cluster is
stationary (the gravitational potential does not change on a sound crossing
time), forces other than gas pressure and gravity (magnetic fields, for
example) are not important, and any motions in the gas are subsonic.
Estimates of intracluster magnetic fields based on radio observations
(Chapter 4) show that they are much too
small to have a significant
dynamical effect. In Section 5.6, we will
show that supersonic expansion of
the gas (cluster winds) is unlikely. Some clusters show evidence for a
slow settling of the intracluster gas due to cooling at the cluster center
(a cooling flow; Sections 4.3,
4.4, and 5.7),
but these motions are very
subsonic except possibly very near the cluster center (*r*
1 kpc; equation
5.105).

Under these circumstances, the gas obeys the hydrostatic equation
(5.56), and
the cluster mass *m*(*r*) can be determined if the density and
temperature of the
intracluster gas are known. This method of determining the mass has a number
of advantages over the use of the virial theorem
(Section 2.8) or any other
method which uses the galaxies as test particles. First, the gas is a
collisional fluid, and the particle velocities are isotropically
distributed. On the other hand, galaxies in clusters (or stars in
galaxies) are collisionless, and uncertainties in the velocity
anisotropy can significantly affect mass determinations. Second, the
hydrostatic method gives the mass as a function of radius, rather than
the total
mass alone as given by the virial method. Third, the statistical accuracy
of this method is not limited by the number of galaxies in the cluster;
the statistical accuracy can be improved by lengthening the observation
time. Fourth, better statistics in the X-ray measurements means that
it is easier to avoid problems with background contamination, and to
resolve possible uncertainities due to subclustering
(Geller and Beers,
1982).
Finally, hydrostatic mass determinations are not very sensitive to
the shape of the cluster
(Strimpel and Binney,
1979;
Fabricant *et al.*,
1984).

The first applications of this method to determine mass distributions
were by
Bahcall and Sarazin
(1977)
and Mathews (1978).
The method has been developed extensively by
Fabricant *et al.* (1980,
1984)
and Fabricant and
Gorenstein (1983).
Applications of the method have been reviewed by
Sarazin (1986b).
Ideally, one would measure the spatially and spectrally resolved X-ray
surface brightness
*I*_{}(*b*)
to directly deconvolve the gas density and temperature
as a function of the radius
(Section 5.5.4; equations 5.80 through
5.83). The mass is then given by the hydrostatic equation, which can be
written as

Note that the mass depends only weakly on the gas density
*n*_{e} (only its
logarithmic derivative enters), but depends strongly on the gas temperature.

Unfortunately, as discussed in
Section 5.5.4,
the limited spectral response of
the *Einstein* X-ray observatory has prevented the direct
determination of temperature profiles for the intracluster gas. Accurate
profiles of the gas density are known. In order to apply the hydrostatic
method to clusters, some simple
assumption must be made about the temperature distribution
*T*_{g}(*r*).
Unfortunately, because the mass is strongly affected by
*T*_{g}, this means that the
resulting mass profiles will be very uncertain. Several analyses
(Vallee, 1981;
Fabricant *et al.*,
1984)
have assumed that gas is isothermal
(Section 5.5.1) or that
the gas temperature and density are related by a simple polytropic equation
(Section 5.5.2).
These analyses give a somewhat smaller total cluster mass than
previous virial estimates, and as noted in
Section 5.5.1, somewhat higher gas masses.

As discussed in Section 4.3,
excellent global cluster X-ray spectra exist from
the HEAO-1 A-2 detectors. These spectra generally cannot be fit by
emission at a single temperature
(Henriksen, 1985).
The spectra can be
used to determine the amount of gas (or, more precisely, the amount of
*EI* = *n*_{p} *n*_{e}*V*, where
*V*is the volume (equation 4.3)) as a function of
temperature, but cannot tell us where the gas is located because of
their poor spatial resolution. The *Einstein* imaging observations
give *n*_{e}(*r*) (which
can be integrated to give *n*_{p}
*n*_{e}*V*), but give no information about the
temperature structure. However, the comparison of these two results
((*EI* vs. *T*_{g}) and (*EI* vs. *r*))
allow the determination of (*T*_{g} vs. *r*), if *we
assume that T _{g} is a monotonic function of the radius r*
(Henriksen, 1985;
Henriksen and Mushotzky,
1986;
Cowie

If these monotonic temperature mass determinations are correct, one is lead to a different picture of the missing mass than that presented in Section 2.8. If the missing mass is more concentrated than the visible matter in the cluster, it suggests that the missing mass has undergone dissipation. Combined with the smaller ratio of missing mass to visible mass, this suggests that the missing mass is baryonic matter, and not some weakly interacting species. The major uncertainty in these analyses is the assumption of a monotonic temperature gradient. While this seems quite plausible, there is no compelling physical argument requiring that this be true. While most detailed models for the ejection or infall of intracluster gas give monotonic gradients (Section 5.10), some do not.

When spatially and spectrally resolved X-ray surface brightness measurement are available, it will be possible to directly determine the mass profiles of clusters of galaxies and determine the distribution of the missing mass. The same method can be applied to the X-ray emitting gas in elliptical galaxies (Section 5.8). This capability should become available with the launch of AXAF (Chapter 6). Until that time, the present results on the distribution of the missing mass must be regarded as tantalizing but tentative.